GMAT 数学题（8）

sdcar2010

请问 ( 5⁹⁹⁹ + 6⁹⁹⁹ ) 的最小质因数是多少？为什么？

sdcar2010

这道题还没想出来，还想问一题有点擦边的
100！+1 的最大质因子是多少？ 谢谢

-- by 会员 小困蛇 (2011/1/4 15:49:59)

No two consecutive positive integers(n, n+1) are ever divisible by same number except 1.
So all the factors of 100! is not a factor of 100!

100! = 100*99*98*97 . . . *50*49*48* . . . .*4*3*2*1

Its prime factors include all the prime numbers smaller than 100.

So the smallest prime factor for (100! + 1) is bigger than 100.  So it might be 101! As to the biggest prime factor of (100! + 1), I have no idea.

sdcar2010

When n is an odd number,

(aⁿ + bⁿ) = (a+b)*(a^n-1 – a^n-2*b + a^n-3*b² – a^n-4*b³ + . . . – a²*b^n-3 + a*b^n-2 – b^n-1 )

So (a+b) is a factor of (aⁿ + bⁿ ) when n is odd.

sdcar2010

是11

(a+b)^n-a^n-b^n必然有(a+b)的因子
根据 杨辉三角

-- by 会员 心中的永恒 (2011/1/4 15:36:47)

Only if n is an odd number.

Right answer.

小困蛇

这道题还没想出来，还想问一题有点擦边的
100！+1 的最大质因子是多少？ 谢谢

心中的永恒

是11

(a+b)^n-a^n-b^n必然有(a+b)的因子
根据 杨辉三角

cherry12345

I think 999 is an odd number, since 11 is the common factor of 5 and 6, so it could be devided by 11, or the sum of 5 and 6.  I am not sure if this is right, as 7 and 13 cannot work.

sdcar2010

What about 11? As 11 is the least common factor of 5 and 6.

-- by 会员 cherry12345 (2011/1/4 13:01:13)

I am curious about 11 as well.  However, how can you prove that the sum of ( 5⁹⁹⁹ + 6⁹⁹⁹ )  can be devided by 11, or the sum of 5 and 6???

cherry12345

What about 11? As 11 is the least common factor of 5 and 6.

sdcar2010

By now, you propably can figure out that 5 is not a factor, either, since 6^n cannot be devided by 5.

What about 7?  

What about 11?

What about 13?

And why the above numbers can or cannot be the factor of the sum?